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Mirrors > Home > MPE Home > Th. List > psrvscacl | Structured version Visualization version GIF version |
Description: Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrvscacl.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrvscacl.n | ⊢ · = ( ·𝑠 ‘𝑆) |
psrvscacl.k | ⊢ 𝐾 = (Base‘𝑅) |
psrvscacl.b | ⊢ 𝐵 = (Base‘𝑆) |
psrvscacl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
psrvscacl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
psrvscacl.y | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
psrvscacl | ⊢ (𝜑 → (𝑋 · 𝐹) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrvscacl.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | psrvscacl.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑅) | |
3 | eqid 2825 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 2, 3 | ringcl 18922 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
5 | 4 | 3expb 1153 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
6 | 1, 5 | sylan 575 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐾) |
7 | psrvscacl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
8 | fconst6g 6335 | . . . . 5 ⊢ (𝑋 ∈ 𝐾 → ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
10 | psrvscacl.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
11 | eqid 2825 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
12 | psrvscacl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
13 | psrvscacl.y | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
14 | 10, 2, 11, 12, 13 | psrelbas 19747 | . . . 4 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
15 | ovex 6942 | . . . . . 6 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
16 | 15 | rabex 5039 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
18 | inidm 4049 | . . . 4 ⊢ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∩ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
19 | 6, 9, 14, 17, 17, 18 | off 7177 | . . 3 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
20 | 2 | fvexi 6451 | . . . 4 ⊢ 𝐾 ∈ V |
21 | 20, 16 | elmap 8156 | . . 3 ⊢ ((({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹) ∈ (𝐾 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹):{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
22 | 19, 21 | sylibr 226 | . 2 ⊢ (𝜑 → (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹) ∈ (𝐾 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
23 | psrvscacl.n | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
24 | 10, 23, 2, 12, 3, 11, 7, 13 | psrvsca 19759 | . 2 ⊢ (𝜑 → (𝑋 · 𝐹) = (({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑋}) ∘𝑓 (.r‘𝑅)𝐹)) |
25 | reldmpsr 19729 | . . . . . 6 ⊢ Rel dom mPwSer | |
26 | 25, 10, 12 | elbasov 16291 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
27 | 13, 26 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
28 | 27 | simpld 490 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
29 | 10, 2, 11, 12, 28 | psrbas 19746 | . 2 ⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
30 | 22, 24, 29 | 3eltr4d 2921 | 1 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 {crab 3121 Vcvv 3414 {csn 4399 × cxp 5344 ◡ccnv 5345 “ cima 5349 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 ∘𝑓 cof 7160 ↑𝑚 cmap 8127 Fincfn 8228 ℕcn 11357 ℕ0cn0 11625 Basecbs 16229 .rcmulr 16313 ·𝑠 cvsca 16316 Ringcrg 18908 mPwSer cmps 19719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-tset 16331 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mgp 18851 df-ring 18910 df-psr 19724 |
This theorem is referenced by: psrlmod 19769 psrass23l 19776 psrass23 19778 mpllsslem 19803 |
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